Department of Mathematical Sciences, Ritsumeikan University Web Page

May 28, 2015 @BKC campus, Forest House F107, 16:30-18:00

1. Time: 16:30-17:10

Freddy Delbaen(ETH and UNIZH)

Title: Risk Measures, Orlicz Spaces and Mackey Topology

Abstract: For a risk measure defined on \(L^\infty\), there is a stronger continuity property which can be translated as (sequential) continuity on the bounded sets for the Mackey topology induced by \(L^1\).  This is a more complicated way of saying that the risk measure satisfies a dominated convergence theorem.  It was proved (by Cheridito and Li) that this implies a continuity property on the whole space (not just on bounded sets).  One can ask whether such a phenomenon exists for convex functions defined on the dual of an arbitrary Banach space.  There are several extensions possible.  The general result, although straightforward in convex function theory,  seems new.  Whether sequential continuity implies continuity is related to a property called strongly weakly compactly generated (SWCG).  There are counterexamples for spaces not satisfying the SWCG property.   In the talk I will put emphasis on the relation between risk measures, Orlicz spaces and Mackey topology.  I will not make the excursion to Banach space theory.

2. Time: 17:20-18:00

André Martinez(University of Bologna and Ritsumeikan University)

Title: Optimal estimates for Helmholtz resonators with straight neck in any dimension.

Abstract: This is a joint work with Alain Grigis and Thomas Duykaerts where we obtain an optimal bound on the width of the lowest resonance for a general Helmholtz resonators with straight neck. Such a resonator consists of a bounded cavity, connected with the exterior by a thin tube. The frequency of the sounds it produces are determined by the shape of the cavity, and one expects that their duration is related to the length of the tube and to the diameter of its section. From a mathematical point of view, this phenomena is described by the resonances of the Dirichlet Laplacian on the domain consisting of the union of the cavity, the tube, and the exterior. Here, we obtain an asymptotic estimate on the duration of the sounds when the width of the tube tends to zero. This estimate shows that this duration is exponentially large, with a rate that is proportional to the length of the tube, and inversely proportional to its width. The proof is based on previous results obtained by Hislop-Martinez and Martinez-Nédélec, and on a recent version of Carleman estimates due to Kenig-Sjostrand-Uhlman.